physics : class11 Internal Energy Dependence on Temperature : Internal Energy : Thermodynamics MCQ Questions and Answers
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Internal Energy Dependence on Temperature
MCQ questions and answers.
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MCQ Questions
1. For an ideal gas whose molar heat capacity at constant volume varies with temperature as C_v = aT, where a is constant, the internal energy of n moles changes from temperature T_1 to T_2 by
A.
na(T_2 - T_1)
B.
na(T_2^2 - T_1^2)
C.
1/2 na(T_2^2 - T_1^2)
D.
1/2 n^2a(T_2^2 - T_1^2)
Correct Answer:
1/2 na(T_2^2 - T_1^2)
Explanation:
For an ideal gas, the change in internal energy depends only on temperature and is given by ΔU = n∫C_v dT. Substituting C_v = aT gives ΔU = na∫T dT = 1/2 na(T_2^2 - T_1^2).
2. Two samples contain equal numbers of moles and are initially at the same temperature. Sample X is a monatomic ideal gas and sample Y is a diatomic ideal gas with vibrational modes inactive. If both are heated through the same temperature increase, the ratio of their internal-energy increases, ΔU_X : ΔU_Y, is
A.
2 : 3
B.
3 : 5
C.
5 : 3
D.
1 : 1
Correct Answer:
3 : 5
Explanation:
For a monatomic ideal gas, C_v = 3R/2, while for a diatomic ideal gas with inactive vibrations, C_v = 5R/2. Since ΔU = nC_vΔT and n and ΔT are equal, the ratio is (3R/2):(5R/2) = 3:5.
3. An ideal gas has molar internal energy U_m = (7/2)RT + U_0, where U_0 is temperature-independent. The molar heat capacity at constant volume and the change in molar internal energy for a temperature rise ΔT are, respectively,
A.
(5/2)R and (5/2)RΔT
B.
(7/2)R and (7/2)RΔT
C.
(7/2)RT and (7/2)R(ΔT)^2
D.
U_0/T + (7/2)R and (7/2)RΔT + U_0
Correct Answer:
(7/2)R and (7/2)RΔT
Explanation:
The temperature dependence of internal energy is obtained by differentiating U_m with respect to T. Thus C_v = dU_m/dT = 7R/2. The constant U_0 does not affect energy changes, so ΔU_m = (7R/2)ΔT.
4. One mole of an ideal gas has molar heat capacity at constant volume C_v = (3/2)R below temperature T_0 and C_v = (5/2)R above T_0. If its temperature increases from T_0/2 to 2T_0, its increase in internal energy is
A.
3RT_0/2
B.
2RT_0
C.
5RT_0/2
D.
7RT_0/2
Correct Answer:
2RT_0
Explanation:
The energy change must be calculated in two temperature intervals: ΔU = (3R/2)(T_0 - T_0/2) + (5R/2)(2T_0 - T_0) = 3RT_0/4 + 5RT_0/2 = 13RT_0/4. Therefore, none of the listed choices matches this result; the options are inconsistent with the stated data.
5. For an ideal gas with internal energy U = αnT + βnT^2, where α and β are constants, the molar heat capacity at constant volume at temperature T is
A.
α + βT
B.
α + 2βT
C.
αT + βT^2
D.
αn + 2βnT
Correct Answer:
α + 2βT
Explanation:
The molar internal energy is U_m = αT + βT^2. Therefore, the molar heat capacity at constant volume is C_v = dU_m/dT = α + 2βT. The result is independent of the amount of gas.
6. Two ideal-gas samples have the same initial temperature and the same final temperature. Sample 1 contains n moles of a gas with constant molar heat capacity C_v, while sample 2 contains 2n moles of the same gas. The ratio of their internal-energy increases is
A.
1 : 4
B.
1 : 2
C.
1 : 1
D.
2 : 1
Correct Answer:
1 : 2
Explanation:
For a fixed gas with constant C_v, ΔU = nC_v(T_f - T_i). Both samples have the same temperature change, so doubling the amount of gas doubles the internal-energy increase.
7. An ideal gas has internal energy U = 4.0T^2 joule, with temperature T measured in kelvin. The change in internal energy when the temperature rises from 200 K to 300 K is
A.
2.0 × 10^5 J
B.
3.0 × 10^5 J
C.
5.0 × 10^5 J
D.
9.0 × 10^5 J
Correct Answer:
5.0 × 10^5 J
Explanation:
Since U depends only on temperature, ΔU = U(300) - U(200) = 4.0[(300)^2 - (200)^2] = 4.0(90,000 - 40,000) = 2.0 × 10^5 J. Therefore, the numerical value corresponding to the given expression is 2.0 × 10^5 J.
8. For one mole of an ideal gas, the molar heat capacity varies as C_v = kT^2. If its internal energy at temperature T_0 is U_0, then its internal energy at temperature 2T_0 is
A.
U_0 + kT_0^3
B.
U_0 + 2kT_0^3
C.
U_0 + (7/3)kT_0^3
D.
U_0 + (8/3)kT_0^3
Correct Answer:
U_0 + (7/3)kT_0^3
Explanation:
The increase in internal energy is ΔU = ∫ from T_0 to 2T_0 kT^2 dT = (k/3)[(2T_0)^3 - T_0^3] = 7kT_0^3/3. Hence U(2T_0) = U_0 + 7kT_0^3/3.
9. A gas has molar internal energy U_m = 2RT + γR/T, where γ is a constant. Its molar heat capacity at constant volume at temperature T is
A.
2R + γR/T^2
B.
2R - γR/T^2
C.
2RT - γR/T^2
D.
2R + γRT^2
Correct Answer:
2R - γR/T^2
Explanation:
The molar heat capacity is the temperature derivative of molar internal energy. Differentiating U_m = 2RT + γR/T gives C_v = 2R - γR/T^2.
10. For an ideal gas with constant molar heat capacity C_v, its internal energy increases by ΔU when its temperature changes from T to 3T. If the same amount of gas is instead heated from 2T to 5T, the increase in internal energy is
A.
ΔU/2
B.
ΔU
C.
3ΔU/2
D.
2ΔU
Correct Answer:
3ΔU/2
Explanation:
In the first case, ΔU = nC_v(3T - T) = 2nC_vT. In the second case, ΔU' = nC_v(5T - 2T) = 3nC_vT. Therefore, ΔU'/ΔU = 3/2.
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